On the dual of (non)-weakly regular bent functions and self-dual bent functions

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November 2013, 7(4): 425-440. doi: 10.3934/amc.2013.7.425

On the dual of (non)-weakly regular bent functions and self-dual bent functions

Ayça Çeşmelioǧlu ^1, , Wilfried Meidl ^2, and Alexander Pott ^1,

1.	Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany, Germany
2.	MDBF, Sabanci University, Orhanlı, Tuzla 34956, İstanbul, Turkey

Received July 2012 Revised March 2013 Published October 2013

Abstract
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For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of non-weakly regular bent functions and show conditions under which their dual is bent as well. This leads to the definition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a different algebraic degree.

Keywords: Fourier transform., self-dual bent functions, Duals of bent functions.

Mathematics Subject Classification: Primary: 11T71, 94A60, 06E30; Secondary: 11T2.

Citation: Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425

References:

[1]	C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28.
[2]	C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.
[3]	C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399. doi: 10.1504/IJICOT.2010.032864.
[4]	C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681.
[5]	A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429. doi: 10.1016/j.jcta.2011.10.002.
[6]	A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242. doi: 10.1007/s10623-012-9686-2.
[7]	Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266. doi: 10.1007/s10801-010-0270-4.
[8]	Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103.
[9]	K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090. doi: 10.1016/j.ffa.2008.07.002.
[10]	F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203. doi: 10.1007/978-3-642-15874-2_16.
[11]	T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854.
[12]	T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130.
[13]	T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291. doi: 10.1007/s12095-011-0048-0.
[14]	X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198. doi: 10.1007/s10623-011-9544-7.
[15]	K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013. doi: 10.1109/TIT.2010.2050806.
[16]	N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[17]	R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997.
[18]	Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003.
[19]	Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159.

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References:

[1]	C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28.
[2]	C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.
[3]	C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399. doi: 10.1504/IJICOT.2010.032864.
[4]	C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681.
[5]	A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429. doi: 10.1016/j.jcta.2011.10.002.
[6]	A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242. doi: 10.1007/s10623-012-9686-2.
[7]	Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266. doi: 10.1007/s10801-010-0270-4.
[8]	Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103.
[9]	K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090. doi: 10.1016/j.ffa.2008.07.002.
[10]	F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203. doi: 10.1007/978-3-642-15874-2_16.
[11]	T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854.
[12]	T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130.
[13]	T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291. doi: 10.1007/s12095-011-0048-0.
[14]	X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198. doi: 10.1007/s10623-011-9544-7.
[15]	K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013. doi: 10.1109/TIT.2010.2050806.
[16]	N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[17]	R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997.
[18]	Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003.
[19]	Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159.

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